Anisotropy of microstructure and elastic properties of niobium carbide nanopowders

Solid State Sciences 100 (2020) 106092

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Anisotropy of microstructure and elastic properties of niobium carbide nanopowders Aleksandr I. Gusev Institute of Solid State Chemistry, Ural Branch of the Russian Academy of Sciences, Pervomaiskaya str. 91, Ekaterinburg, 620990, Russia

A R T I C L E I N F O

A B S T R A C T

Keywords: Niobium carbides High-energy ball milling Neutron diffraction Nanocrystalline powders Size inhomogeneity Elastic moduli and microstrain anisotropy

The peculiarities of microstructure of nanocrystalline powders of nonstoichiometric niobium carbides have been generalized in all homogeneity interval of cubic NbCy phase. Nanocrystalline NbCy powders were produced by high-energy milling of coarse-crystalline powders. For the first time, it is established that the microstructure of nanocrystalline NbCy powders is an inhomogeneous, and contains two cubic fractions differing by particle size and carbon content. Fine details of an inhomogeneous microstructure of NbCy nanopowders have been deter­ mined by the time-of-flight neutron high resolution Fourier diffractometry. The analysis of the neutron diffraction data revealed microstrain anisotropy in the nanopowders. The average sizes of the coherent scattering regions and microstrains in nanocrystallites with allowance for the anisotropy of the deformation distortions have been estimated. The consideration of the anisotropy of elastic properties has shown that the splitting of niobium carbide particles during high-energy milling is likely to take place in the (110) or (111) planes in the directions where the shear modulus has the least value.

1. Introduction In modern engineering, cubic MCy (M ¼ Ti, Zr, Hf, V, Nb, Ta, y � 1.0) carbides are used in the manufacture of tool materials capable of oper­ ating at high temperatures, in aggressive environments, and under high loads [1]. Niobium carbide NbCy with a cubic B1-type crystal structure and a broad homogeneity interval from NbC0.70 to NbC1.00 holds a special place among the nonstoichiometric carbides. The properties of the NbCy carbide depend on carbon content, y, and vary strongly across its homogeneity interval. Niobium carbide is one of the most refractory material and one of the hardest of all known substances: the melting temperature of the nonstoichiometric NbC0.80 carbide is equal to 3880 K, and the microhardness HV of the NbC0.77 carbide is 33.0 GPa [1,2]. Unique feature of nonstoichiometric cubic carbides is the formation of ordered phases M2tC2t-1 with t ¼ 1, 1.5, 2, 3, and 4 [1,3]. According to Refs. [1,3–5], ordered monoclinic (space group C2/m) Nb6C5 phase is formed in nonstoichiometric NbCy in the composition range from NbC0.81 to NbC0.88 at a temperature below 1300 K. According to the data of [1], a formation of second ordered phase Nb3C2 is possible at a temperature below 900 K in the homogeneity interval of cubic NbCy carbide along with the main ordered phase Nb6C5. Fine nonstoichiometric carbides powders are used in the manufac­ ture of nanostructured hardmetals [6], as well as doping impurities

capable of ensuring precipitation hardening of heat-resistant steels [7–9]. According to Ref. [8], when alloying cast iron and steel with vanadium or niobium carbides, nanoscale dispersed carbide particles are released in the metal matrix in the form of a monoclinic ordered phase. The application of niobium carbide as a component of a cutting tool for machining metals and wear protection is discussed in detail in recent review articles [10,11]. According to Refs. [10,11], niobium carbide as a lightweight material than tungsten carbide WC shows a very high potential for many technical applications. Ball milling is effective approach for the producing different nano­ crystalline materials including carbide powders [12–14]. Preparing nanocrystalline nonstoichiometric niobium carbide NbCy powders is accompanied by microstructure change. High-energy ball milling of nonstoichiometric carbides was studied in detail in works [6,15–17]. It was shown theoretically and experi­ mentally that the particle size of nanopowders prepared by milling of nonstoichiometric carbides depends on carbide nonstoichiometry. The effect of nonstoichiometry and small particle size on the microstructure features of niobium and vanadium carbide nanopowders prepared by high-energy milling was revealed also [18–22]. In recent decades, intensive investigations of mechanical alloying of metal alloys have been carried. Results of high-energy milling of metal alloys are described in the monograph [14] and a large number of

E-mail address: [email protected]. https://doi.org/10.1016/j.solidstatesciences.2019.106092 Received 30 September 2019; Received in revised form 3 December 2019; Accepted 4 December 2019 Available online 9 December 2019 1293-2558/© 2019 Elsevier Masson SAS. All rights reserved.

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Solid State Sciences 100 (2020) 106092

original articles. In particular, the mechanical milling of Ti–Nb and Nb–Al alloys was studied by the authors [23,24]. In contrast to carbides which are brittle and very hard materials, a feature of the milling of ductile (plastic) metal alloys is the formation of amorphous phases. Microstructure of milled carbide nanopowders is due to their elastic and strength characteristics and deformation mechanisms at splitting of coarse-crystalline particles. When a powder is milled, the energy is expended on the breaking of interatomic bonds in a crystal and on the creation of the additional surfaces resulting from the splitting of crys­ talline particles. The splitting of the cubic carbides and the motion of dislocations occurs mainly by the slip systems f111g〈110〉 and f110g〈110〉 [25–27], i. e. along the (111) or (110) planes in the ½110� direction (Fig. 1). Slippage of close-packed atomic planes in the ½110� direction corresponds to the Burgers vector b with the length b ¼ aB1√2/2. For cubic niobium and tantalum carbides, it is preferable to slip along the (110) planes in the ½110� direction [28]. Hardness anisotropy measurements in niobium carbide have also shown slip to occur on both the (111) and (110) planes; however, TEM evidence has only reported (111) slip [27–29]. At temperatures below 1000 K, carbides are brittle materials. Their brittleness is not associated with a spasmodic change in the mechanism responsible for plastic deformation, but is due to the fact that the stress necessary for the movement of a dislocation is so great that it leads to fracture. Several research groups have recently identified that transition metal carbides become tougher as its y ¼ C/M ratio drops from 1.0 to about 0.7 [27,28,30]. The increase in melting temperature when the composition MCy carbides deviates from stoichiometry, is considered as an indication of a stronger covalent bonding. This phenomenon has been reported by several computational and experimental studies [31–33]. In other words, metallic bonding is increased and covalent bonding is decreased with growth y ¼ C/M in niobium and tantalum carbides. Thus, stoichiometric carbides are less brittle than non-stoichiometric carbides. According to the calculations [28–30,34–36], the f110g〈110〉 slip system has the lowest shear strength under pure shear deformation. This means that it is the most likely active slip system during deformation. The fracture mechanism along the f110g〈110〉 slip system arises due to the stretching and breaking of the metal-carbon bonds with increasing shear strain. For a biaxial shear, the f110g〈110〉 slip system has the lowest critical shear stress. The main parameters of the microstructure of milled carbide nano­ powders are nanoparticle size D and microstrain ε. The theoretical dependence of the particle size D of milled powder on

the mass M and the particle size Din of the initial coarse-crystalline powder, duration t and the energy of milling, Emill, was derived in Refs. [16,17] taking into account the nonstoichiometry of cubic carbides MCy and features of their deformation. This dependence has the form: � � M Ap ðyÞ þ Bp ðyÞεðy; t; MÞln½Din =2bðyÞ� �� : � Dðy; t; MÞ ¼ (1) Emill ðtÞ þ M Ap ðyÞ þ Bp ðyÞεðy; t; MÞln½Din =2bðyÞ� Din In formula (1), Emill(t) ~ kt is the energy of milling proportional to the duration of milling (the coefficient k depends on the design of the milling plant and the mechanics of movement of grinding bodies); ε is microstrains appearing in the substance during milling; b ¼ |b| is the modulus of the Burgers vector; Ap and Bp are certain constants typical of a given carbide MCy and depending on its properties (density dd, shear modulus G, energy of interatomic bonds u, etc.) and carbon content y; in particular, constant B is proportional to Gb/(1-μ) where μ is Poisson ratio. The dependences of NbCy properties on the carbide composition are given in Refs. [1,16,37]. Formula (1) satisfies the edge condition D (0,M) ¼ Din since in the initial time t ¼ 0 the energy of milling Emill(0) ¼ 0 and microstrains ε(0,M) ¼ 0. The duration of milling, t, is the most important parameter. Usually the time t is chosen so as to achieve a stable equilibrium between the fracture and agglomeration of powder particles. The study of nanocrystalline powders of nonstoichiometric carbides requires an accurate assessment of their crystal structure, possible ordering, phase composition, shape and size of nanoparticles (coherent scattering regions (CSR)), magnitude and spatial distribution of micro­ strains, type and quantity of other defects. The listed characteristics of nanostructured substances are analyzed using fast, reliable, nondestructive and unbiased methods [12]. Exact microstructure attesta­ tion is very important in order to utilize nanocrystalline non­ stoichiometric carbides. Among the available techniques for obtaining this information in a non-destructive way, the leading role belongs to diffraction methods [38]. Diffraction of the short-wave radiation (X-ray or neutron) is one of the methods for studying nanocrystalline compounds: It can provide information on the size of particles and microstrains from the broad­ ening of the diffraction reflections. The most widely used and available tools for the study of nanocrystalline substances and compounds are conventional X-ray diffractometers, which use monochromatic radiation with a fixed wavelength. Much less often, neutron diffractometers with a fixed constant wavelength are used to study the microstructure of nanocrystalline substances. One of the important advantages of neutron diffraction in comparison with X-ray diffraction is the high sensitivity to

Fig. 1. The slip systems (a) f111g〈110〉 and (b) f111g〈110〉 in nonstoichiometric carbides MCy with the basis B1 type cubic structure. The f111g〈110〉 slip system has the lowest critical shear stress under biaxial shear deformation, and the f111g〈110〉 slip system has the lowest shear strength under pure shear deformation. b ¼ ½110�/2 is the Burgers vector showing the direction of slipping in the (111) and (110) planes. 2

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light elements. In the case of nonstoichiometric carbides, the intensities of scattering of neutrons by the nuclei of atoms of transition metals and carbon are comparable in magnitude, which makes it possible to detect the ordering of the light carbon atoms. At the end of the twentieth century, powerful pulsed neutron sources IBR-2 (Joint Institute of Nuclear Research, Dubna, USSR), SNS (Oak Ridge National Laboratory, Oak Ridge, USA), ISIS (Rutherford Appleton Laboratory, Chilton, Oxfordshire, England), and JSNS (J-PARS Center, Tokai, Ibaraki, Japan) were built. On the basis of these sources, time-offlight (TOF) neutron diffractometers were created. These diffractome­ ters are not inferior to conventional diffractometers on stationary re­ actors, but even surpass them in many characteristics. A distinctive feature of modern TOF-diffractometers for polycrystals is a high resolution. TOF-diffractometers on pulse neutron sources are the most prom­ ising neutron instruments for structure and microstructure studies of nanocrystalline substances including nonstoichiometric carbides. The usage of a continuous neutron spectrum makes it possible to cover a very wide interval of interplanar distances dhkl from ~0.05 to ~1.5 nm and more. In addition, neutrons from a pulsed source are slowed down to thermal energies and their speed is small, so it is possible to analyze quantitatively the neutrons wavelength by time of flight. It is also important that the resolution of the TOF diffractometer depends very weakly on dhkl and improves with growth d. These factors make it possible to estimate the effect of different defects of the crystal on the profile and reflection width more accurately than on a conventional diffractometer with a monochromatic neutron beam. This investigation is devoted to generalizing the microstructure pe­ culiarities of nanocrystalline NbCy (y ¼ 0.77, 0.84, 0.93, 0.96) powders produced by high-energy ball milling of coarse-crystalline powders of nonstoichiometric cubic niobium carbides NbC0.77, NbC0.84, NbC0.93, and NbC0.96. Peculiarities of a microstructure of nanocrystalline pow­ ders are first summarized in the entire homogeneity interval of cubic niobium carbide. A set of X-ray and neutron diffraction techniques along with other experimental and calculated methods has been used to investigate the microstructure of nanopowders of nonstoichiometric niobium carbides NbCy.

performed on a Shimadzu XRD-7000 X-ray diffractometer in CuKα1, 2 radiation. The X-ray diffraction (XRD) patterns were analyzed using the X’Pert HighScore Plus [41]. The average size D of particles (more pre­ cisely, the average size of coherent scattering region (CSR)) in milled niobium carbide powders was determined from the diffraction reflection broadening. All the diffraction reflections were described by the pseudo-Voigt function. The average particle size D was also estimated from the value of specific surface area Ssp measured by the Brunauer-Emmett-Teller (BET) method. The neutron experimental data were obtained using the IBR-2 (Joint Institute of Nuclear Research, Dubna) neutron source and also SINQ (Paul Scherrer Institute, Villigen) neutron source. High-resolution neutron diffraction patterns of all samples were recorded at a temperature of 293 K on an HRFD high-resolution time-offlight Fourier diffractometer [42] installed at the channel 5 of the long-pulse IBR-2 reactor (Joint Institute of Nuclear Research, Dubna). The studied powders were placed in vanadium containers. The instru­ ment employs the correlation technique of data acquisition, which provides a very high resolution (Δd/d � 0.0013 at d ¼ 0.2 nm). This resolution should guarantee the possibility of determining the micro­ strain in the domains at the level of ε � 5 � 10 4, and the average size of coherently scattering domains of Dcoh � 350 nm. NIST standard refer­ ence material SRM 676 (Al2O3) was used for the determination of the resolution function WR of the HRFD diffractometer. The initial pro­ cessing of the data array measured on a TOF diffractometer using a continuous neutron spectrum was carried out with a scan on interplanar distance taking into account all instrumental corrections using the MRIA software package [43]. Then neutron diffraction patterns were analyzed using WPPM method [44,45] implemented in PM2K software program package. Additionally, neutron diffraction measurements of coarse- and nanocrystalline NbC0.84 powders were performed at the Paul Scherrer Institute (PSI, Villigen, Switzerland) on a HRPT diffractometer [46] using a monochromatic thermal neutron beam at wavelength of 0.11545 nm. 3. Results and discussion

2. Samples and experimental methods

Primary XRD structural attestation has shown that the initial coarsecrystalline NbCy (0.77, 0.84, 0.93, 0.96) powders contain one cubic (space group Fm3m) phase with B1-type crystal structure. The lattice constant a of the initial powders varies from 0.44377 nm for NbC0.77 to 0.44648 nm for NbC0.96. High-energy milling of coarse-crystalline NbCy powders led to X-ray diffraction reflections broadening [16] (see Fig. S1, Supplementary Material) caused by the small size D of CSR and the presence of microstrains ε. The quantitative analysis of XRD reflections broadening has shown that the size of CSR varies from ~50 to ~20 nm with an increase in the milling duration from 5 to 15 h. Coarse-crystalline powder consists of large particles of more 1–2 μm along with small particles about 0.5 μm in size, and does not contain aggregates. According to SEM data, the average particle size of NbC0.93 nanopowders is several times larger than the CSR size estimated from XRD data. This indicates agglomeration of nanoparticles. Indeed explicit agglomeration is observed in nanopowders. In the powder produced by high-energy ball milling for 5 h, small particles about 200 nm in size are combined into large agglomerates with a size of up to 5 μm [17,18]. At duration of milling t ¼ 15 h, smaller particles of size ~100 nm form large agglomerates to 4–5 μm in size also. Neutron diffraction patterns (λ ¼ 0.11545 nm) of the coarsecrystalline NbC0.84 powder and the same powder after ball milling for 10 h obtained on a HRPT (PSI) diffractometer are shown in Fig. 2. Highenergy milling led to an increase in the width of diffraction reflections. Refinement of neutron diffraction patterns confirmed the cubic (space group Fm3m) structure carbide powders. The occupancy y of 4(b) sites of

The initial coarse-crystalline niobium carbides NbCy (y ¼ 0.77, 0.84, 0.93, 0.96) powders with an average particle size of 2–3 μm were pro­ duced by patented high-temperature vacuum sintering of niobium powder with calcined carbon black at a maximal temperature of 2000 K [39]; the sintering technique is described in detail elsewhere [1] (see experimental details in Supplementary Material). This is the most reli­ able method of synthesis that ensures the preparing carbide samples NbCy with a specified carbon content y. Other, even the most modern methods of synthesis (for example, carbothermal reduction of niobium oxide Nb2O5 by melamine C3N3(NH2)3 in vacuum at ~1300 K [40]) make it possible to prepare only stoichiometric or close to stoichiometric NbC~1.0 niobium carbide. The nanocrystalline niobium carbides NbCy powders were produced by high-energy ball milling in a PM-200 Retsch planetary ball mill with the angular speed of rotation ω equal to 8.33 revolutions per second (rps) for 5, 10, and 15 h. The mass of powder taken for milling was 10 g. The milling technique is described in detail in works [6,15]. For a PM-200 Retsch planetary ball mill with the angular speed of rotation ω ¼ 8.33 rps, the energy of milling is Emill(t) ¼ 0.781t [J] [16] (see also Supplementary Material). Production and attestation of nanocrystalline niobium carbide powders are described in detail in Refs. [16,17]. The morphology, particle size and elemental chemical composition of the initial and milled niobium carbide powders were examined by the scanning electron microscopy (SEM) method on a JEOL-JSM LA 6390 microscope coupled with a JED 2300 Energy Dispersive X-ray Analyzer. The primary structural characterization of NbCy powders was 3

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Fig. 3. Model neutron diffraction pattern (λ ¼ 0.11545 nm) of NbC0.84 carbide ordered as monoclinic (space group C2/m) Nb6C5 superstructure. Calculation was performed in the approximation of maximum long-range order parameter ηmax ¼ 0.96. The inset presents the part of calculated neutron diffraction pattern with diffraction reflections of monoclinic ordered phase of the Nb6C5 type. Long and short ticks mark the positions of the diffraction reflections of disordered cubic (space group Fm3m) NbC0.84 phase of the B1 type and ordered monoclinic (space group C2/m) phase of the Nb6C5 type, respectively.

Neutron diffraction pattern of the ordered NbC0.84carbide, calculated in approximation of the maximum long-range order with ηmax ¼ 0.96, is shown in Fig. 3. For a monochromatic beam of neutrons with wave­ length 0.11545 nm, the set of specified superstructure reflections of monoclinic phase of Nb6C5 type should be observed in the region of 2θ < 24� (Fig. 3). Such reflections are absent on experimental neutron diffraction patterns (see Fig. 2). Thus, coarse- and nanocrystalline NbC0.84 carbides contain only disordered cubic phases. Diffraction measurements have shown that NbC0.77, NbC0.93, and NbC0.96 carbides are disordered too. Fig. 4 shows the summarized neutron diffraction patterns of the initial coarse-crystalline NbCy powders and NbCy nanopowders recorded on the HRFD diffractometer by using the time-of-flight (TOF) mode of data acquisition. Part of these spectra has been presented earlier in studies [18–21]. A disordered cubic (space group Fm3m) carbide phase is the main phase of powders studied. No any superstructure reflections of the ordered phases are present on the neutron diffraction patterns of NbCy powders. The diffraction reflections of nanopowders are strongly broadened because of milling; furthermore, the diffuse background in­ creases. Very narrow peaks observed on diffraction patterns near d ~0.123, ~0.152, and ~0.214 nm are reflections from vanadium containers. Broad peaks on neutron diffraction patterns of NbCy nano­ powders near d ~0.185–0.190 nm are (011) reflections of impurity nanocrystalline hexagonal (space group P6m2) tungsten carbide WC. The WC impurity appears during milling because of rubbing of grinding balls and inner shell of bowls made of the hard WC alloy with 6 wt % Co. For example, the nanopowder produced by 15-h milling of NbC0.84 carbide contains 96.1 wt % niobium carbide and 3.9 wt % WC. The presence of 2.1–4.7 wt % WC as an impurity phase was detected in all of the NbCy nanopowders produced by high-energy milling.

Fig. 2. Neutron diffraction patterns (λ ¼ 0.11545 nm) of (a) coarse-crystalline and (b) nanocrystalline NbC0.93 powders: (£) experiment, ( ) calculation. The difference (Iobs - Icalc) between the experimental and calculated neutron diffraction patterns is shown in the lower part of the figure, the position of the Bragg reflections is indicated by ticks. Long vertical ticks mark the positions of impurity hexagonal tungsten carbide WC. The measurements were performed on a HRPT diffractometer (PSI, Villigen, Switzerland).

the non-metallic face centered cubic sublattice with carbon atoms C is 0.84(3). This agrees with the results of chemical and XRD analysis of NbC0.84 carbide. 3.1. Possibility of ordering Composition of NbC0.84 carbide is very close to the composition of perfect ordered monoclinic (space group C2/m) Nb6C5 (NbC0.833) phase [1,4]. In the case of ordering of NbC0.84 carbide, superstructure re­ flections of monoclinic phase should be observed in neutron diffraction pattern. According to Ref. [1], dependence of the maximum value of the long-range order parameter on the composition of nonstoichiometric compound MXy at formation of superstructure M2tX2t 1 type has the form � 2tð1 yÞ; if y � ð2t 1Þ=2t; (2) ηmax ðyÞ ¼ 2ty=ð2t 1Þ; if y < ð2t 1Þ=2t:

3.2. Elastic properties The particles of nanocrystalline NbCy powders are monocrystalline, therefore the anisotropy of elastic properties of niobium carbide leads to the anisotropy of nanoparticle microstructure. In the elasticity theory, the anisotropic Young modulus Ehkl and the Poisson ratio μhkl of cubic

For NbC0.84 carbide at the formation of Nb6C5 (t ¼ 3) superstructure, maximum value of the long-range order parameter, ηmax, is 0.96. 4

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Solid State Sciences 100 (2020) 106092

Fig. 4. HRFD neutron diffraction patterns of the initial coarse-crystalline NbCy powder and NbCy nanopowders produced by milling for 5, 10, and 15 h. Experimental ( � ) and calculated ( ) neutron diffraction patterns are shown. On neutron diffraction patterns of nanopowders, the upper and lower rows of vertical ticks mark the calculated positions of the diffraction reflections for nanocrystalline F1 and coarse-crystalline F2 fractions, respectively. Partially reprinted from Ref. [20] with permission from Wiley.

crystals depend on the direction [hkl] and are determined through the elastic stiffness constants c11, c12 and c44 or the components of compli­ ance tensor (elastic compliance constants) s11, s12 and s44 [47,48] as

Ehkl ¼ s11

� 2 s11

2 2

1 1 s 2 44

s12

2 2

� ; μhkl ¼ 1 = 2 Γ

Ehkl 1 � 2ðc11 þ 2c12 Þ

nanoparticle size and high-energy milling parameters it follows that the shear modulus G is the main elastic characteristic affecting the nano­ particle size. Fig. 5 displays the calculated dependences of the shear modulus Ghkl and the Poisson ratio μhkl of cubic niobium carbide NbC1.0.

Ehkl ðs11 þ 2s12 Þ � 2

Considering (3), the shear modulus Ghkl is equal to. 2s12

6ðs11

s12

s44 = 2ÞΓ�

1 s 2 44

� Γ �;

(3)

1 s 2 44

For the calculation we used the rounded values of the components of the compliance tensor s11 ¼ 1.8⋅10 12, s12 ¼ 0.3⋅10 12 and s44 ¼ 7.0⋅10 12 Па 1, corresponding to data [49], which are most close to the experi­ mental elastic characteristics of niobium carbide generalized in works [17,37]. The distributions of the shear modulus Ghkl are shown in the (100), (110) and (111) planes (Fig. 5a,b,c). The distribution of the Poisson ratio μhk0 of cubic niobium carbide is shown in the (100) plane (Fig. 5d). The spatial distribution of the shear modulus Ghkl of cubic niobium carbide is demonstrated in Fig. 6. The shear modulus Ghkl has the maximal value ~238 GPa in [100], [010], [001] and inverse directions and the minimal value ~143–150 GPa in directions [11l], where l changes from 0 to ~0.6 (see also Fig. 5b). The value of shear modulus Ghkl in (111) plane is small, almost independent of the direction and varies from ~146 to ~159 GPa (Fig. 5c). It is clear that during milling of niobium carbide the particles are cleaved along the (110) or (111)

2 2

l þk l where Γ ¼ h ðhk 2þh is the anisotropy factor for cubic crystals. þk2 þl2 Þ2

Ghkl ¼ 1 = ½2s11

� s12 þ s11 s12 � s11 2 s11 s12

(4)

The elastic stiffness constants c11, c12, c44 and the elastic compliance constants s11, s12, s44 for cubic crystals are related by the known re­ lations: s44 ¼ 1/c44, s11 ¼ (c11þc12)/[(c11-c12)(c11þ2c12)] and s12 ¼ -c12/ [(c11-c12)(c11þ2c12)]. In the literature, the information about the elastic characteristics c11, c12 and c44 of niobium carbide, calculated by the generalized gradient approximation (GGA) or the local density approximation (LDA), are presented in works [49–53]. The data on s11, s12 and s44 for niobium carbide make it possible to find the distributions of elastic characteristics of monocrystalline par­ ticles of cubic NbC from direction [hkl]. From relation (1) between the 5

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Solid State Sciences 100 (2020) 106092

Fig. 5. The dependences of elastic characteristics of monocrystalline NbC particles on the crystallographic direction: distributions of the shear modulus Ghkl in (a) plane (100), (b) plane (110) and (c) plane (111); distributions of (d) Poisson ratio μhk0 of cubic niobium carbide in plane (100).

planes in the directions [11l] with a small l, where the shear modulus has the least value. This agrees with the conclusions of studies [25–27] that the splitting of niobium carbide particles and motion of dislocations during milling are most likely to occur by the slip system f110g〈110〉 or f110g〈110〉, i. е. along the (110) or (111) planes in the ½110� direction. 3.3. Analysis of diffraction reflections broadening The main origins of the broadening of diffraction reflections are the small particle size D and microstrains ε of the crystal lattice [38] owing to its deformation distortions and atomic displacements because of the presence of dislocations [54]. In general, dislocation, stacking faults, and sub-boundaries defects cause strains, but can also affect grain size determination. The software packages used for processing TOF neutron diffraction patterns account for and separate all the contributions related to the broadening of diffraction reflections, and exclude the in­ fluence of these defects on the determination of particle size. The par­ ticle size D is determined by extrapolating the dependence between the reduced broadening [β(2θ)cosθ]/λ and the scattering vector s ¼ (2sinθ)/λ to the value s ¼ 0. Experimentally, this is achieved by simul­ taneously scanning of the reciprocal space along the wavelength λ and the wave vector of neutron after their scattering. With this in mind, the dependence of the diffraction reflection width

Fig. 6. The spatial distribution of the shear modulus Ghkl of cubic niobium carbide. 6

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Solid State Sciences 100 (2020) 106092

β 2 of these reflections, which is less than 3 � 10 6 nm2. Therefore in the case of nanocrystalline powders, it is possible to accurately determine the broadening β 2 of only diffraction reflections with even indices (hkl) in the region of d > 0.09 nm. The quantitative description of the HRFD neutron diffraction pat­ terns (see Fig. 4) has shown that NbCy nanopowders contain dominant nanocrystalline powder fraction along with small quantity of coarsecrystalline fraction. The dependences β 2(d2) were determined from the HRFD spectra with considering two fractions in NbCy nanopowders. These dependences β 2(d2) are shown in Fig. S2 (see Supplementary Material) and are well described by the function (6) with a correction for the anisotropy factor. At the same milling duration t, the broadening β 2 increases with an increasing carbon content y in NbCy carbide. This is due to the fact that the contributions of size and strain broadening to the general broad­ ening of diffraction reflections depend on changes in the mechanical properties (elastic moduli, fracture stress, compression strength, etc.) of NbCy carbide in the homogeneity interval [16]. The quantitative analysis of the neutron diffraction patterns for NbCy nanopowders revealed that the diffraction reflections are best described in the approximation of two cubic (space group Fm3m) powder fractions with different average particle sizes and different lattice constants which correspond to different compositions of carbide. It is very important result. Each of the observed diffraction reflections is the sum of narrow and broad peaks. The broad and narrow peaks correspond to the nanosized fraction F1 and coarse-crystalline fraction F2, respectively. The composition of NbCy nanoparticles was determined by the value of the crystal lattice constant aB1 found by the pseudo-Voigt function fitting of broadened diffraction reflections. The dependence of the crystal lattice constant on the composition (relative carbon content, y) of disordered NbCy carbide was determined by us earlier (see our mono­ graph [1]) with very high accuracy and is described by the function aB1(y) ¼ 0.003778|(y 0.80445)|1/3 þ 0.44469 � 0.00005 [nm]. The accuracy of determining the composition of NbCy nanoparticles by this method is y�0.004. The dependences of the lattice constant aB1 and the relative carbon content y for NbCy nanoparticles on the milling duration t are shown in Fig. 7. As an example, the expansion of the (200)B1 diffraction reflections of the NbC0.77 and NbC0.96 nanopowders into peaks corresponding to different cubic carbide fractions F1 and F2 are shown in Fig. 8. Both nanopowders are produced by milling for 5 h. It is seen that total reflection of each nanopowder describes by two contributions. The main contribution (>93%) to reflection is made by the broad peak corre­ sponding to the nanosized fraction F1. Broad peak is displaced into the region of smaller d about 0.0004 nm relative to the narrow peak that

Δd or measured value W ¼ KΔd on the polycrystal interplanar distance d during measurement on an HRFD diffractometer is described [19] as W 2 � ðKΔdÞ2 ¼ C1 þ C2 d2 þ C3 ðA þ BΓÞd2 þ C4 d4

(5)

where W ¼ K⋅Δd is measured reflection width (K ¼ 25912.7 is the constant for the HRFD diffractometer at measuring Δd in nm), W2R ¼ C1 þ C2 d2 is the resolution function, and C3(A þ BГ )d2 and C4d4 are the contributions from the strain and size broadening. According to Refs. [18,55,56], an additional dislocation-related anisotropy factor Γ that is responsible for the anisotropy of microstrains, is included in the strain broadening contribution of eq. (5). The coefficients C1 and C2 are related to the resolution function and the HRFD diffractometer parameters. The coefficients C3 and C4 have the form C3 ¼ (2kρ)2K2 and C4 ¼ (khkl/D)2K2, ε is microstrain, and D is the size of CSR. The coefficient kρ is the con­ stant depending on the density of dislocations and the Burgers vector b, i. e. on the variation of the interplanar distance and atomic displace­ ments. As already noted, the motion of dislocations in cubic niobium carbides occur mainly by the slip system f110g〈110〉, and this slippage corresponds to the Burgers vector with the length b ¼ aB1√2/2. The coefficient (A þ BΓ) takes into account the edge and screw dislocations in the deformed crystal, and A and B are the constants for the given sample depending on the dislocation density and their relative content. Considering the resolution function WR, the experimental broad­ ening β 2 ¼ W2 W2R of arbitrary reflection (hkl) has the form β2hkl ¼ C3 ðA þ BΓÞd2hkl þ C4 d4hkl :

(6)

Thus, the microstrain taking into account the anisotropy of crystal

deformation is εhkl ¼ 12½C3 ðA þ BΓÞ�1=2 . The consideration of the value and anisotropy of microstrains is particularly important in strongly deformed materials [55,57–59]. Discussed carbide nanopowders are such strongly deformed materials. The intensities of diffraction reflections (hkl) are proportional to Phkl F2hkl , where Phkl is a multiplicity factor and F2hkl is a structural factor. For cubic niobium carbide NbCy, the structural factors of diffraction reflections with all even and all odd indices (hkl) are equal to 16(fNb þ yfC)2 and 16(fNb-yfC)2, where fNb ¼ 0.71 � 10 12 and fC ¼ 0.665 � 10 12 cm are the coherent scattering lengths of neutrons by niobium and carbon nuclei [60]. Taking into account the values of fNb and fC, the intensities of diffraction reflections with odd indices (hkl) are many times smaller than the intensities of reflections with even indices in the neutron experiment on niobium carbide (see Fig. 4). The error in determining the broadening of reflections located at d < 0.09 nm is ~ (1.5–2.5)∙10 6 nm2, and is comparable in magnitude to the broadening

Fig. 7. Effect of milling duration, t, on the characteristics of the nanosized fraction of NbCy powders: (a) lattice constant aB1 and (b) relative carbon content y in niobium carbide nanoparticles. 7

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Solid State Sciences 100 (2020) 106092

Fig. 8. Expansion of the (200)B1 diffraction reflections into peaks corresponding to cubic fractions F1 and F2 of niobium carbide with different lattice constant, composition, and average particle size: (a) NbC0.77 nanopowder produced by milling for 5 h, and (b) NbC0.96 nanopowder produced by milling for 5 h. The experimental points and calculated curves are shown. Partially reprinted from Ref. [20] with permission from Wiley.

corresponds to fraction F2. Thus, two cubic fractions F1 and F2 with different lattice constants aB1 and different particle size co-exist in the NbCy nanopowders. The lattice constant of the nanosized fraction F1 for all nanopowders decreases weakly at an increase in the milling duration t, whereas the lattice constant of the coarse-crystalline fraction F2 is almost indepen­ dent of the duration of milling and is ~0.44389, ~0.44576, ~0.44642 and ~0.44661 nm for NbC0.77, NbC0.84, NbC0.93, and NbC0.96 nano­ powders, respectively. The detection of two fractions with different particle sizes in niobium carbide nanopowders can be useful for producing the most compact (non-porous) hardmetals. If the furnace mixture is monodispersed, then the volume cannot be filled without voids with particles of equal size. In practice, more compact packing is achieved due to the presence of different-sized particles in the furnace mixture, when the finer particles occupy the voids between the coarser particles. Study of the threedimensional packing of spherical particles [61] has shown that a very high degree of the free space filling can be reached in a case of bimodal

particle size distribution when, along with large particles there is a large number of small particles. Earlier, similar results on a micro-nonuniform structure have been received for nanopowders of vanadium carbide [22], tungsten carbide [62], titanium monoxide [63,64] and of lead and silver sulfides [65–68]. Nonstoichiometry takes place in these sulfides too but its value is much less than in cubic vanadium and niobium carbides and titanium monoxide. It should be noted that conventional XRD study of different milled powders including NbCy and TaCy nanopowders could not detect their inhomogeneous fractional composition [16,17]. The dependences of nanoparticle size D and average microstrains (microdeformations) εaver on the milling duration, t, and NbCy compo­ sition, y, are shown in Fig. 9. The size D of nanoparticles (more precisely, the size of coherent scattering regions) for all niobium carbide nanosized fractions decreases with an increase in the duration of milling t (Fig. 9a). Nanoparticles have the smallest size D in NbC0.96 carbide nanopowders, the closest to the

Fig. 9. Effect of milling duration, t, and NbCy composition, y, on (a) the size of coherent scattering region, D, of the nanosized fraction F1 of niobium carbide powders, and (b) the average microstrains εaver: (●) experimental nanoparticle size and average microstrain determined from high-resolution neutron diffraction patterns. Model surface Dtheor(y,t) is calculated by formula (1). 8

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Solid State Sciences 100 (2020) 106092

stoichiometric composition. Model surface Dtheor(y,t) (see Fig. 9a) is calculated by formula (1) with the mass M of the initial powder 10 g, the initial particle size Din ¼ 5 μm, and milling energy Emill(t) ¼ 0.781t [J]. Good agreement between model dependence Dtheor(y,t) and the experi­ mental size D of nanopowders obtained during 5, 10 and 15 h milling of NbC0.77, NbC0.84, NbC0.93 and NbC0.96 carbides is observed (Fig. 9a). Milling of cubic niobium carbide of any composition within its ho­ mogeneity interval leads to a rapid decrease of the particle size to ~100 nm under during 1500–2000 s, and to ~70 nm during a 1-h milling. Further increase of the milling duration to 10–15 h or more is accom­ panied by a slow asymptotic reduction of the particle size to a certain constant value (see Fig. 9a). An increase in the milling duration over 15 h (20, 25, 30 h or more) does not lead to an increase in the size of NbCy nanoparticles, since niobium carbide is a brittle and very solid sub­ stance. The change in particle size of carbide powders from the milling duration is described in detail in articles [15–17], as well as in our monograph [6]. Growth of the nanoparticles size with an increase in the milling time t above a certain critical value tcrit can be observed for ductile (plastic) substances (metals and alloys). The values of microstrains εaver grow with increase in the milling duration t, reaching ~0.9–1.2% for the nanopowders produced by milling for 15 h (Fig. 9b). Increasing carbon content, y, in NbCy carbides is accompanied by weak decrease of microstrains εaver. The smoothed surface εaver(y,t) is constructed on experimental εaver data. The increase in the milling duration t is accompanied by an increase in the content of the nanosized fraction F1 from ~93% at t ¼ 5 h to 99–100% at t ¼ 15 h, and decreasing the content of coarse-crystalline fraction F2 to 0 at maximal milling duration t ¼ 15 h. According to SEM data, particle size of coarse-crystalline fraction F2 is equal to ~150–200 nm and does not depend on the milling duration. As the milling duration t is increased, the content of the nanocrystalline fraction F1 grows, while the concentration of the coarse-crystalline fraction F2 becomes smaller. From the dependence between the lattice constant of NbCy and the relative carbon content y [1], it follows that the nano- and coarse-crystalline fractions have different compositions. For example, fractions F1 and F2 of NbC0.77 powder have the compositions ~ NbC0.73-0.76 and ~NbC0.79, respectively; the same fractions of NbC0.84 powders have the compositions ~ NbC0.81-0.82 and ~NbC0.83-0.84, etc. The approximation of the experimental data β 2(d2) for NbCy

nanopowders by function (6) allows to determine the εhkl microstrains. Fig. 10 shows the distributions of the microstrains εhkl over the di­ rections [hkl] for the NbC0.77 and NbC0.96 nanopowders produced by 10 and 15 h milling, respectively. The radii of the spheres are proportional to the microstrain εaver averaged over all crystallographic directions, and the lengths of the vectors are proportional to the εhkl values. The average value of the εaver microstrains was found in accordance with [18] as P P εaver ¼ ð εhkl Phkl Þ= Phkl , i.e. by averaging the εhkl taking into consideration the multiplicity factor Phkl. It is seen from Fig. 10a that microstrains ε220, ε222, and ε422, in the [220], [222], and [422] directions are 0.68%, 0.76%, and 0.77%, respectively, and less than εaver � 0.81%. Microstrains ε200 and ε420 in the [200] and [420] directions are equal to 0.96% and 0.84%, and exceed εaver. In the NbC0.96 nanopowder prepared by 15 h milling, the microstrains ε200, ε220, ε222, ε420, and ε422 are equal to 1.25%, 1.19%, 1.35%, 1.27%, and 1.27%, respectively, at εaver � 1.27% (see Fig. 10b). Direction [220] is equivalent to direction [110], i.e. coincides with preferable slip direction of close-packed atomic planes in cubic carbides [30–36]. The most likely active slip system during deformation is slip­ page in the [110] direction. Found distributions of microstrains εhkl correspond to a slippage in the [110] � [220] direction: For NbC0.77 and NbC0.96 nanopowders, the microstrains ε220 have the least relative values (see Fig. 10). Niobium carbide nanopowders studied have an inhomogeneous fractional composition, i.e., they have some size distribution. Standard X-ray and neutron diffraction methods using monochromatic radiation cannot determine details of such distribution. However, these details can be revealed on the HRFD neutron diffractometer using a continuous neutron radiation. It follows from the results obtained that a careful control of the milling conditions is necessary in order to produce a nanocrystalline nonstoichiometric carbide powders with a uniform size distribution and a desired composition. On the one hand, the homogeneity of the microstructure of niobium carbide nanopowders is a prerequisite for their use as grain growth inhibitors in hardmetals. Moreover, fine ho­ mogeneous niobium carbide powders can enhance solid-state densifi­ cation during hardmetals sintering. On the other hand, the bimodal size distribution of particles contributes to a more complete filling of the volume during sintering of nanopowders and producing more dense

Fig. 10. Distributions of microstrains εhkl over the nonequivalent [hkl] directions: (a) NbC0.77 nanopowder prepared by 10 h milling, εaver ¼ 0.0081 (or 0.81%), (b) NbC0.96 nanopowder prepared by 15 h milling, εaver ¼ 0.0127 (or 1.27%). The radii of the spheres are proportional to the εaver, the vectors length in the [hkl] directions is proportional to εhkl microstrains. 9

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Solid State Sciences 100 (2020) 106092

alloys and hardmetals. Thus, controlling the high-energy ball milling conditions, especially the milling duration, it is possible to prepare nanopowders containing one or two fractions with different particle sizes.

[7] G.L. Dunlop, D.A. Porter, Secondary precipitation of ordered V6C5 and (V, Ti)6C5 particles in ferrite, Scand. J. Metall. 6 (1977) 19–20. [8] R. Kesri, S. Hamar-Thibault, Structures ordonnees a longue distance dans les carbures MC dans les fonts, Acta 36 (1988) 149–166. [9] S.G. Huang, R.L. Liu, O. van der Biest, J. Vleugels, NbC as grain growth inhibitor and carbide in WC–Co hardmetals, Intern. J. Refr. Met. Hard Mater. 26 (2008) 389–395. [10] M. Woydt, S. Huang, J. Vleugels, H. Mohrbacher, E. Cannizza, Potentials of niobium carbide (NbC) as cutting tools and for wear protection, Intern. J. Refr. Met. Hard Mater. 72 (2018) 380–387. [11] M. Woydt, S. Huang, E. Cannizza, J. Vleugels, H. Mohrbacher, Niobium carbide for machining and wear protection - evolution of properties, Metal Powd, For. Rep. 74 (2019) 82–89. [12] A.I. Gusev, A.A. Rempel, Nanocrystalline Materials, Cambridge Intern. Science Publ., Cambridge, 2004, p. 351. [13] H.-E. Schaefer, Nanoscience. The Science of the Small in Physics, Engineering, Chemistry, Biology and Medicine, Springer, Heidelberg - Dordrecht - New York, 2010, p. 772. [14] P. Bal� a�z, Mechanochemistry in Nanoscience and Minerals Engineering, Springer, Berlin – Heidelberg, 2008, p. 413. [15] A.I. Gusev, A.S. Kurlov, Production of nanocrystalline powders by high-energy ball milling: model and experiment, Nanotechnology 19 (2008) 8, paper 265302. [16] A.S. Kurlov, A.I. Gusev, Effect of nonstoichiometry of NbCy and TaCy powders on their high-energy ball milling, Intern. J. Refr. Met. Hard Mater. 46 (2014) 125–136. [17] A.S. Kurlov, A.I. Gusev, High-energy milling of nonstoichiometric carbides: effect of nonstoichiometry on particle size of nanopowders, J. Alloy. Comp. 582 (2014) 108–118. [18] A.S. Kurlov, I.A. Bobrikov, A.M. Balagurov, A.I. Gusev, Neutron diffraction study of nanocrystalline NbC0.93 powders and the anisotropy of deformation distortions, JETP Lett. (Engl. Transl.) 100 (2014) 629–634. [19] A.M. Balagurov, I.A. Bobrikov, G.D. Bokuchava, R.N. Vasin, A.I. Gusev, A. S. Kurlov, M. Leoni, High-resolution neutron diffraction study of microstructural changes in nanocrystalline ball-milled niobium carbide NbC0.93, Mater. Char. 109 (2015) 173–180. [20] A.S. Kurlov, A.I. Gusev, A.A. Rempel, V.S. Kuznetsov, I.A. Bobrikov, A. M. Balagurov, Evolution of microstructure of niobium carbide NbC0.77 powders, Cryst. Res. Technol. 52 (2017) 7, paper 1700061. [21] A.S. Kurlov, V.S. Kuznetsov, I.A. Bobrikov, A.M. Balagurov, A.I. Gusev, Neutron diffraction study of microstructure of nanocrystalline cubic niobium carbides from NbC0.77 to NbC0.96, Asian Res. J. Curr. Sci. 1 (2018) 10, paper ARJOCS.61. [22] A.S. Kurlov, V.S. Kuznetsov, I.A. Bobrikov, A.M. Balagurov, A.I. Gusev, A. A. Rempel, Microinhomogeneity of the structure of nanocrystalline niobium and vanadium carbides, JETP Lett. (Engl. Transl.) 108 (2018) 253–259. [23] C. Salvo, C. Aguilar, R. Cardoso-Gil, A. Medina, L. Bejar, R.V. Mangalaraja, Study on the microstructural evolution of Ti-Nb based alloy obtained by high-energy ball milling, J. Alloy. Comp. 720 (2017) 254–263. [24] X. Li, X. Wen, H. Zhao, Z. Ma, L. Yu, C. Li, C. Liu, Q. Guo, Y. Liu, The formation and evolution mechanism of amorphous layer surrounding Nb nano-grains in Nb-Al system during mechanical alloying process, J. Alloy. Comp. 779 (2019) 175–182. [25] R. Viswanadham, Science of Hard Materials, Springer, Berlin - Heidelberg, 2012, p. 1012. [26] D.J. Rowcliffe, G.E. Hollox, Hardness anisotropy, deformation mechanisms, and brittle-to-ductile transition in carbides, J. Mater. Sci. 6 (1971) 1270–1276. [27] R.H.J. Hannink, D.L. Kohlstedt, M.J. Murray, Slip system determination in cubic carbides by hardness anisotropy, Proc. R. Soc. Lond. A 326 (1972) 409–420. [28] H. Yu, M. Bahadori, G.B. Thompson, C.R. Weinberger, Understanding dislocation slip in stoichiometric rocksalt transition metal carbides and nitrides, J. Mater. Sci. 52 (2017) 6235–6248. [29] E. Bitzek, J.R. Kermode, P. Gumbsch, Atomistic aspects of fracture, Int. J. Fract. 191 (2015) 13–30. [30] K. Hackett, S. Verhoef, R.A. Cutler, D.K. Shetty, Phase constitution and mechanical properties of carbides in the Ta–C system, J. Am. Ceram. Soc. 92 (2009) 2404–2407. [31] Z. Wu, X.-J. Chen, V.V. Struzhkin, R.E. Cohen, Trends in elasticity and electronic structure of transition-metal nitrides and carbides from first principles, Phys. Rev. B 71 (2005) 5, paper 214103. [32] H. Li, L. Zhang, Q. Zeng, K. Guan, Structural, elastic and electronic properties of transition metal carbides TMC (TM¼ Ti, Zr, Hf and Ta) from first-principles calculations, Solid State Commun. 151 (2011) 602–606. [33] J. Yang, F. Gao, First principles calculations of mechanical properties of cubic 5d transition metal monocarbides, Phys. B Condens. Matter 407 (2012) 3527–3534. [34] N. de Leon, X.X. Yu, H. Yu, C.R. Weinberger, G.B. Thompson, Bonding effects on the slip differences in the B1 monocarbides, Phys. Rev. Lett. 114 (2015) 5, paper 165502. [35] C.R. Weinberger, X.X. Yu, H. Yu, G.B. Thompson, Ab initio investigations of the phase stability in group IVB and VB transition metal nitrides, Comput. Mater. Sci. 138 (2017) 333–345. [36] B. Tang, Y. Shen, Q. An, Shear-induced brittle failure of titanium carbide from quantum mechanics simulations, J. Am. Ceram. Soc. 101 (2018) 4184–4192. [37] M.G. Di Vernieri Cuppari, S.F. Santos, Physical properties of the NbC carbide, Metals 6 (2016) 17, paper 250. [38] E.J. Mittemeijer, P. Scardi (Eds.), Diffraction Analysis of the Microstructure of Materials, Springer, Berlin, 2004, p. 554. [39] L.-M. Berger, M. Hermann, A.I. Gusev, A.A. Rempel, Verfahren zur Herstellung � nichtstAchiometrischer Carbide definierter Zusammensetzung. Offenlegungsshrift

4. Conclusion The NbCy nanopowders that produced by high-energy ball milling have not a fully homogeneous fractional composition and contain two cubic fractions with different particle size, i.e. these nanopowders have a bimodal size distribution. The standard diffraction methods based on using monochromatic radiation do not allow one to determine the de­ tails of such distribution, which however were established by combined analysis of the profile of all diffraction reflections of niobium carbide nanopowders measured on an HRFD neutron diffractometer. All the characteristics (background, shape and broadening of diffraction re­ flections, lattice constant and composition of nanoparticles, size of particles and values of microstrains, content of nanosize and coarsecrystalline fractions, etc) are determined by the high-accuracy refine­ ment of experimental TOF neutron diffraction patterns. A comprehen­ sive analysis of the neutron diffraction data revealed microstrain anisotropy in the nanopowders. The calculation of elastic properties and the determination of microstrains in niobium carbide have shown that the most probable slip system in the process of high-energy milling of NbCy carbides is slippage along (110) or (111) planes in the ½110� direction. The study of the niobium carbide nanopowders has generally shown that the time-of-flight neutron diffraction technique is promising for studying strongly deformed nonstoichiometric compounds. It is shown that TOF neutron diffraction patterns can provide the very accurate data for microstructure analysis of nanocrystalline powders. Declaration of competing interest The author declares no conflict of interest. Acknowledgements Author is obliged to Dr. A.S. Kurlov for the ball milling of coarsecrystalline NbCy powders. Author is grateful to Prof. A.M. Balagurov and Dr. I.A. Bobrikov for the experimental HRFD measurements, and to Dr. V. Yu. Pomjakushin and Dr. D.V. Sheptyakov for the the experiment on HRPT diffractometer. Appendix A. Supplementary data Supplementary data to this article can be found online at https://doi. org/10.1016/j.solidstatesciences.2019.106092. References [1] A.I. Gusev, A.A. Rempel, A.J. Magerl, Disorder and Order in Strongly Nonstoichiometric Compounds. Transition Metal Carbides, Nitrides and Oxides, Springer, Berlin – Heidelberg – New York – London, 2001, p. 608, https://doi.org/ 10.1007/978-3-662-04582-4. [2] W. Huang, Thermodynamic evaluation of Nb C system, Mater. Sci. Technol. 6 (1990) 687–694. [3] C.H. de Novion, B. Beuneu, T. Priem, N. Lorenzelli, A. Finel, Defect structures and order-disorder transformations in transition metal carbides and nitrides, in: R. Freer (Ed.), The Physics and Chemistry of Carbides, Nitrides and Borides, Kluwer Acad. Publ., Netherlands, 1990, pp. 329–355. [4] A.I. Gusev, A.A. Rempel, Order-disorder phase transition channel in niobium carbide, Phys. stat. sol.(a) 93 (1986) 71–80. [5] J.P. Landesman, A.N. Christensen, C.H. de Novion, N. Lorenzelli, P. Convert, Orderdisorder transition and structure of the ordered vacancy compound Nb6C5: powder neutron diffraction studies, J. Phys. C Solid State Phys. 18 (1985) 809–823. [6] A.S. Kurlov, A.I. Gusev, Tungsten Carbides: Structure, Properties and Application in Hardmetals, Springer, Cham-Heidelberg-New York-Dordrecht-London, 2013, p. 256.

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[40] [41] [42] [43] [44] [45] [46]

[47]

[48] [49] [50] [51]

Solid State Sciences 100 (2020) 106092

DE 198 07 589 A 1, Bundesrepublik Deutschland: Deutschland Patentamt (1998) 1–3. M. Lei, H.Z. Zhao, H. Yang, B. Song, W.H. Tang, Synthesis of transition metal carbide nanoparticles through melamine and metal oxides, J. Eur. Ceram. Soc. 28 (2008) 1671–1677. X’Pert HighScore Plus, Version 2.2e (2.2.5). ©, PANalytical B. V. Almedo, the Netherlands, 2009. A.M. Balagurov, High-resolution Fourier diffraction at the IBR-2 reactor, Neutron News 16 (2005) 8–12. V.B. Zlokazov, V.V. Chernyshev, Mria - a program for a full profile analysis of powder multiphase neutron-diffraction time-of-flight (direct and Fourier) spectra, J. Appl. Crystallogr. 25 (1992) 447–451. P. Scardi, M. Leoni, Whole powder pattern modelling, Acta Crystallogr. A 58 (2002) 190–200. M. Leoni, P. Scardi, Nanocrystalline domain size distributions from powder diffraction data, J. Appl. Crystallogr. 37 (2004) 629–634. P. Fischer, G. Frey, M. Koch, M. K€ onnecke, V. Pomjakushin, J. Schefer, R. Thut, N. Schlumpf, R. Bürge, U. Greuter, S. Bondt, E. Berruyer, High-resolution powder diffractometer HRPT for thermal neutrons at SINQ, Physica B 276–278 (2000) 146–147. T. Gn€ aupel-Herold, P.C. Brand, H.J. Prask, Accessing the elastic properties of cubic materials with diffraction methods, in: Advances in X-Ray Analysis, Proc. Of the 47th Denver X-Ray Conference, Colorado Springs, August 3-7, 1998), vol. 42, ICDD, 1998, pp. 464–470. T. Gn€ aupel-Herold, P.C. Brand, H.J. Prask, Calculation of single-crystal elastic constants for cubic crystal symmetry from powder diffraction data, J. Appl. Crystallogr. 31 (1998) 929–935. A. Zaoui, B. Bouhafs, P. Ruterana, First-principles calculations on the electronic structure of TiCxN1 x, ZrxNb1 xC and HfCxN1 x alloys, Mater. Chem. Phys. 91 (2005) 108–115. W.-X. Feng, S.-X. Cui, H.-Q. Hu, G.-Q. Zhang, Z.-T. Lv, Electronic structure and elastic constants of TiCxN1-x, ZrxNb1-xC and HfCxN1-x alloys: a first-principles study, Physica B 406 (2011) 3631–3635. V. Krasnenko, M.G. Brik, First-principles calculations of hydrostatic pressure effects on the structural, elastic and thermodynamic properties of cubic monocarbides XC (X ¼ Ti, V, Cr, Nb, Mo, Hf), Solid State Sci. 14 (2012) 1431–1444.

[52] L. Wu, Y. Wang, Z. Yan, J. Zhang, F. Xiao, B. Liao, The phase stability and mechanical properties of Nb–C system: using first-principles calculations and nanoindentation, J. Alloy. Comp. 561 (2013) 220–227. [53] D. Hong, W. Zeng, Fu-S. Liu, B. Tang, Qi-J. Liu, Structural, electronic, elastic and mechanical properties of NbC-based compounds: first-principles calculations, Physica B 558 (2019) 100–108. [54] E.J. Mittemeijer, U. Welzel, The “state of the art” of the diffraction analysis of crystallite size and lattice strain, Ztschr. Kristallogr. 223 (2008) 552–560. [55] M. Leoni, J. Martinez-Garcia, P. Scardi, Dislocation effects in powder diffraction, J. Appl. Crystallogr. 40 (2007) 719–724. [56] P. Scardi, M. Ortolani, M. Leoni, WPPM: microstructural analysis beyond the Rietveld method, Mater. Sci. Forum 651 (2010) 155–171. [57] I.C. Dragomir, T. Ung� ar, The dislocations contrast factors of cubic crystals in the Zener constant range between zero and unity, Powder Diffr. 17 (2002) 104–111. [58] I. Lucks, P. Lamparter, E.J. Mittemeijer, An evaluation of methods of diffractionline broadening analysis applied to ball-milled molybdenum, J. Appl. Crystallogr. 37 (2004) 300–311. [59] T. Ung� ar, Dislocation model of strain anisotropy, Powder Diffr. 23 (2008) 125–132. [60] M.T. Hutchings, P.J. Withers, T.M. Holden, T. Lorentzen, Introduction to the Characterization of Residual Stress by Neutron Diffraction, CRC Press, Boca Raton, 2005, p. 420. [61] S.V. Anishchik, N.N. Medvedev, Three-dimensional Apollonian packing as a model for dense granular systems, Phys. Rev. Lett. 75 (1995) 4314–4317. [62] A.S. Kurlov, A.I. Gusev, Phase equilibria in the W – C system and tungsten carbides, Usp. Khim. 75 (2006) 687–708. [63] A.A. Valeeva, A.A. Rempel, A.I. Gusev, Electrical conductivity and magnetic susceptibility of titanium monoxide, JETP Lett. (Engl. Transl.) 73 (2001) 621–625. [64] A.A. Valeeva, K.A. Petrovykh, H. Schroettner, A.A. Rempel, Effect of stoichiometry on the size of titanium monoxide nanoparticles produced by fragmentation, Inorg. Mater. 51 (2015) 1132–1137. [65] S.I. Sadovnikov, N.S. Kozhevnikova, V.G. Pushin, A.A. Rempel, Microstructure of nanocrystalline PbS powders and films, Inorg. Mater. 48 (2012) 21–27. [66] S.I. Sadovnikov, A.I. Gusev, Structure and properties of PbS films, J. Alloy. Comp. 573 (2013) 65–75. [67] S.I. Sadovnikov, A.I. Gusev, Chemical deposition of nanocrystalline lead sulfide powders with controllable particle size, J. Alloy. Comp. 586 (2014) 105–112. [68] S.I. Sadovnikov, A.A. Rempel, Synthesis of nanocrystalline silver sulfide, Inorg. Mater. 51 (2015) 759–766.

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